The “fundamental theorem” for the algebraic K-theory of spaces: I

Let Xmaps toA(X) denote the algebraic K-theory of spaces functor. The main objective of this paper is to show that A(X×S1) admits a functorial splitting. The splitting has four factors: a copy of A(X), a delooped copy of A(X) and two homeomorphic nil terms. One should view the decomposition as the algebraic K-theory of spaces version of the Bass-Heller-Swan theorem. In deducing this splitting, we introduce a new tool: a “non-linear” analogue of the projective line.Elsevier Science B.V.